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<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>AD-cs.out</source>

<heading><![CDATA[
Prover9 (32) version March-2007, March 2007.
Process 20992 was started by mccune on cleo,
Mon Mar 19 17:01:48 2007
The command was "/home/mccune/bin/prover9 -f cs.in AD.in".
]]></heading>

<proof number="1" length="32" max_count="25">

<comments><![CDATA[
% Proof 1 at 0.03 (+ 0.00) seconds: D.
% Length of proof is 32.
% Level of proof is 11.
% Maximum clause weight is 25.
% Given clauses 52.
]]></comments>

  <clause id="1" type="assumption">
    <literal><![CDATA[
      x * y != x * z
    ]]></literal>
    <literal><![CDATA[
      y = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="2" type="assumption">
    <literal><![CDATA[
      x * y != z * y
    ]]></literal>
    <literal><![CDATA[
      x = z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="3" type="assumption">
    <literal><![CDATA[
      (x * y) * z = x * y * z
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="4" type="assumption">
    <literal><![CDATA[
      x * y * (y @ x) = y * x
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="5" type="assumption">
    <literal><![CDATA[
      (x @ y) @ z = u @ (v @ w)
    ]]></literal>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="6" type="assumption">
    <literal><![CDATA[
      (A @ C) * (B @ C) != A * B @ C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[assumption].">
      <j1 rule="assumption"/>
    </justification>
  </clause>

  <clause id="7">
    <literal><![CDATA[
      A * B @ C != (A @ C) * (B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[copy(6),flip(a)].">
      <j1 rule="copy" parents="6"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="8">
    <literal><![CDATA[
      x * y * z != x * y * u
    ]]></literal>
    <literal><![CDATA[
      z = u
    ]]></literal>
    <justification jstring="[para(3(a,1),1(a,1)),rewrite(3(4))].">
      <j1 rule="para" parents="3 1"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="10">
    <literal><![CDATA[
      x * (x @ x) = x
    ]]></literal>
    <justification jstring="[hyper(1,a,4,a)].">
      <j1 rule="hyper" parents="1 4"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * y != z * x
    ]]></literal>
    <literal><![CDATA[
      z * (z @ x) = y
    ]]></literal>
    <justification jstring="[para(4(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="4 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="14">
    <literal><![CDATA[
      x * y * (y @ x) * z = y * x * z
    ]]></literal>
    <justification jstring="[para(4(a,1),3(a,1,1)),rewrite(3(2),3(5)),flip(a)].">
      <j1 rule="para" parents="4 3"/>
      <j2 rule="rewrite" parents="3 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y) * (u @ (v @ w))
    ]]></literal>
    <justification jstring="[para(5(a,1),4(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="5 4"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      (A * B @ C) * x != (A @ C) * (B @ C) * x
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(2,b,7,a),rewrite(3(14))].">
      <j1 rule="ur" parents="2 7"/>
      <j2 rule="rewrite" parents="3"/>
    </justification>
  </clause>

  <clause id="29">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x * (x @ (A * B @ C)) != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[para(4(a,1),21(a,1)),flip(a)].">
      <j1 rule="para" parents="4 21"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      x * y != x
    ]]></literal>
    <literal><![CDATA[
      x @ x = y
    ]]></literal>
    <justification jstring="[para(10(a,1),1(a,1)),flip(a)].">
      <j1 rule="para" parents="10 1"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="39">
    <literal><![CDATA[
      x * (x @ x) * y = x * y
    ]]></literal>
    <justification jstring="[para(10(a,1),3(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 3"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="41">
    <literal><![CDATA[
      (x @ y) * (z @ (u @ v)) = x @ y
    ]]></literal>
    <justification jstring="[para(5(a,1),10(a,1,2))].">
      <j1 rule="para" parents="5 10"/>
    </justification>
  </clause>

  <clause id="45">
    <literal><![CDATA[
      (x @ y) * z = z * (x @ y)
    ]]></literal>
    <justification jstring="[back_rewrite(17),rewrite(41(6))].">
      <j1 rule="back_rewrite" parents="17"/>
      <j2 rule="rewrite" parents="41"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      (x @ x) * y = y
    ]]></literal>
    <justification jstring="[hyper(8,a,39,a)].">
      <j1 rule="hyper" parents="8 39"/>
    </justification>
  </clause>

  <clause id="79">
    <literal><![CDATA[
      x * ((y @ y) @ x) = x
    ]]></literal>
    <justification jstring="[para(70(a,1),4(a,1,2)),rewrite(70(5))].">
      <j1 rule="para" parents="70 4"/>
      <j2 rule="rewrite" parents="70"/>
    </justification>
  </clause>

  <clause id="80">
    <literal><![CDATA[
      x * (x @ (y @ y)) = x * (y @ y)
    ]]></literal>
    <justification jstring="[para(70(a,1),4(a,1))].">
      <j1 rule="para" parents="70 4"/>
    </justification>
  </clause>

  <clause id="105">
    <literal><![CDATA[
      x * (y @ (z @ u)) = x
    ]]></literal>
    <justification jstring="[para(5(a,1),79(a,1,2))].">
      <j1 rule="para" parents="5 79"/>
    </justification>
  </clause>

  <clause id="113">
    <literal><![CDATA[
      x * (y @ y) = x
    ]]></literal>
    <justification jstring="[back_rewrite(80),rewrite(105(3)),flip(a)].">
      <j1 rule="back_rewrite" parents="80"/>
      <j2 rule="rewrite" parents="105"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="115">
    <literal><![CDATA[
      (A @ C) * (B @ C) * x != x * (A * B @ C)
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_rewrite(29),rewrite(105(13))].">
      <j1 rule="back_rewrite" parents="29"/>
      <j2 rule="rewrite" parents="105"/>
    </justification>
  </clause>

  <clause id="116">
    <literal><![CDATA[
      x @ x = y @ y
    ]]></literal>
    <justification jstring="[hyper(36,a,113,a)].">
      <j1 rule="hyper" parents="36 113"/>
    </justification>
  </clause>

  <clause id="120">
    <literal><![CDATA[
      x @ x = c_0
    ]]></literal>
    <justification jstring="[new_symbol(116)].">
      <j1 rule="new_symbol" parents="116"/>
    </justification>
  </clause>

  <clause id="130">
    <literal><![CDATA[
      c_0 * x = x
    ]]></literal>
    <justification jstring="[back_rewrite(70),rewrite(120(1))].">
      <j1 rule="back_rewrite" parents="70"/>
      <j2 rule="rewrite" parents="120"/>
    </justification>
  </clause>

  <clause id="133">
    <literal><![CDATA[
      (x @ y) @ z = c_0
    ]]></literal>
    <justification jstring="[para(120(a,1),5(a,2))].">
      <j1 rule="para" parents="120 5"/>
    </justification>
  </clause>

  <clause id="146">
    <literal><![CDATA[
      C * (A @ C) * A * B * (B @ C) != A * B * C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(11,b,115,a(flip)),rewrite(45(11),3(11),3(18))].">
      <j1 rule="ur" parents="11 115"/>
      <j2 rule="rewrite" parents="45 3 3"/>
    </justification>
  </clause>

  <clause id="153">
    <literal><![CDATA[
      x * C * (A @ C) * A * B * (B @ C) != x * A * B * C
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[ur(1,b,146,a)].">
      <j1 rule="ur" parents="1 146"/>
    </justification>
  </clause>

  <clause id="196">
    <literal><![CDATA[
      x * (y @ x) * y * z = y * x * z
    ]]></literal>
    <justification jstring="[para(14(a,1),14(a,1,2)),rewrite(133(6),130(6))].">
      <j1 rule="para" parents="14 14"/>
      <j2 rule="rewrite" parents="133 130"/>
    </justification>
  </clause>

  <clause id="197">
    <literal><![CDATA[
      $F
    ]]></literal>
    <attribute><![CDATA[
      answer(D)
    ]]></attribute>
    <justification jstring="[back_rewrite(153),rewrite(196(13),4(8)),xx(a)].">
      <j1 rule="back_rewrite" parents="153"/>
      <j2 rule="rewrite" parents="196 4"/>
      <j2 rule="xx"/>
    </justification>
  </clause>

</proof>

</proofs>
